# Provability interpretations of modal logic

@article{Solovay1976ProvabilityIO, title={Provability interpretations of modal logic}, author={Robert Solovay}, journal={Israel Journal of Mathematics}, year={1976}, volume={25}, pages={287-304} }

We consider interpretations of modal logic in Peano arithmetic (P) determined by an assignment of a sentencev* ofP to each propositional variablev. We put (⊥)*=“0 = 1”, (χ → ψ)* = “χ* → ψ*” and let (□ψ)* be a formalization of “ψ)* is a theorem ofP”. We say that a modal formula, χ, isvalid if ψ* is a theorem ofP in each such interpretation. We provide an axiomitization of the class of valid formulae and prove that this class is recursive.

## 422 Citations

Provability, truth, and modal logic

- PhilosophyJ. Philos. Log.
- 1980

Certain systems of (propositional) modal logic that are of special interest in connection with the study of the notions of provability in P(eano) A(arithmetic) and truth in the standard model for PA are discussed.

On the degrees of unsolvability of modal predicate logics of provability

- PhilosophyJournal of Symbolic Logic
- 1994

Vardanyan [11] shows that, provided that MPL(A) will be the set of modal formulas q such that, for every interpretation *, q$* is a theorem of A.

Finite Kripke models and predicate logics of provability

- PhilosophyJournal of Symbolic Logic
- 1990

Abstract The paper proves a predicate version of Solovay's well-known theorem on provability interpretations of modal logic: If a closed modal predicate-logical formula R is not valid in some finite…

Provability as a Modal Operator with the models of PA as the Worlds

- Philosophy
- 2011

This paper introduces a propositional modal model of Gödel-Löb Logic whose worlds are the models of Peano Arithemetic and whose box modality is equivalent to an operator satisfying the Hilbert…

Sequent logic of arithmetic decidability ∗

- Philosophy
- 2016

Our paper continues, on the one hand, the study of modal logics that have arithmetical semantics, and on the other, the investigation of decidability (or “non-contingency”) logics. We present…

On strong provability predicates and the associated modal logics

- Mathematics, PhilosophyJournal of Symbolic Logic
- 1993

Some further modal properties of Dzhaparidze's logic are proved, e.g., the fixed point property and the Craig interpolation lemma and other examples of the strong provability predicates and their applications.

Predicate provability logic with non-modalized quantifiers

- Philosophy, Computer ScienceStud Logica
- 1991

It is proved the recursive enumerability of the sets of Q′-formulas all arithmetical examples of which are: 1) T-provable, 2) true.

Logical equations and admissible rules of inference with parameters in modal provability logics

- Mathematics, PhilosophyStud Logica
- 1990

It is proved that the decision problem for admissibility of rules in Gödel-Löb systemGL and Solovay logicS is decidable and the problem of the existence of finite basis for admissible rules forGL andS has a negative solution.

Bimodal logics for extensions of arithmetical theories

- PhilosophyJournal of Symbolic Logic
- 1996

The bimodal provability logics for certain natural pairs of recursively enumerable theories, mostly related to fragments of arithmetic, are characterized and modal logics with propositional constants with natural Kripke semantics are introduced.

Modal Logic for Induction

- PhilosophyAiML
- 2020

We use modal logic to obtain syntactical, proof-theoretic versions of transﬁnite induction as axioms or rules within an appropriate labelled sequent calculus. While transﬁnite induction proper, also…

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